application of theoretical physics methods in industrial r&d lecture_2012

8/2/2019 Application of Theoretical Physics Methods in Industrial R&D Lecture_2012

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Lecture Notes:

Applications of Methods from Theoretical

Physics in Industry

-

University of Basel

Thomas Christen, ABB Corporate Research

FS 2012


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2

Zeit: Do. 16.00-17.30; abwechselnd Vorlesung mit Ubungen (je alle 2Wochen)

Ubungsassistent: Diego Rainis (Raum 4.6) Ort: Institut fur Physik, Neuer Horsaal 1 Teilnahmevoraussetzung: Bachelor/Vordiplom in Physik o. Nanowissenschaften

o. Mathematik.

Modul: Vertiefungsfach Physik (Master in Physik) Lernziele: Anwendung theoretischer Methoden auf typische Fragestellun-

gen der industriellen F&E.

Hinweise zur Leistungsuberprfung: Aktiv an Ubungen teilgenommen. Geloste

Ubungen nachtrglich dem Assistenten abgegeben (jeweils in der darauffol-genden Ubungsstunde).

Bei Interesse Ende Semester Besuch im ABB Forschungszentrum.


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Contents

1 Economic Modelling 7

1.1 Utility Maximization . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Net Present Value (NPV) . . . . . . . . . . . . . . . . . . . . . . 81.3 Market Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Optimization of Technical Devices 11

2.1 Entropy Production Rate . . . . . . . . . . . . . . . . . . . . . . 112.2 Entropy Generation Minimization . . . . . . . . . . . . . . . . . . 132.3 Endoreversible Thermodynamics . . . . . . . . . . . . . . . . . . 142.4 Power Maximization . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Ragone Plots of Energy Storage Devices . . . . . . . . . . . . . . 192.6 Economic Optimization . . . . . . . . . . . . . . . . . . . . . . . 23

3 Modelling Concepts and Methods 27

3.1 Steady States and Linear Stability Analysis . . . . . . . . . . . . 27

3.2 Collective Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Nonequilibrium Systems . . . . . . . . . . . . . . . . . . . . . . . 353.4 Optimization Principles near Equilibrium . . . . . . . . . . . . . 373.5 Optimization Principles far from Equilibrium . . . . . . . . . . . 42

3


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4 CONTENTS

Theoretical physics in industrial R&D1

Industrial theoretical physics per se, as a counterpart to academic theoreticalphysics, does not exist. Even so, the daily work of a theoretical physicist inindustrial R&D often differs from that in a university institute. 2 This is be-cause industrial R&D is an investment, which has to contribute to a companyseconomic benefit, with the implication of a different prioritization of tasks anda stronger restriction of resources. Three main tasks of industrial R&D are

Optimization of technologies and products. This combines technical andeconomical expertise.

Applied research. Contribution to a basic understanding of technologiesand products. This means to develop models (modelling) for qualitativeand quantitative predictions of the behavior of technical (and sometimes

non-technical) systems.

Innovation. Invention of novel or improved products and/or technologiesby new ideas. Here a deep understanding of physical basics can be a largeadvantage.

For all these tasks, methods of theoretical physics can be useful instruments.The requirement of profitability leads to the following main differences betweenindustrial and university working conditions:

The art of academic physics consists in the isolation of the physical phe-nomenon of interest from all unwanted influences. The well-trained theo-retician reduces the system under consideration to its essential part, whichis mostly a simplification towards a purified but often unrealistic model(e.g., the Hubbart model in solid state physics) that can be treated withrigorous methods.On the other hand, real technical systems are typically exposed to un-avoidable and uncontrolled, sometimes large disturbances, which can leadto a huge complexity. Because the realistic system has to be considered,it must not be further simplified. But the complexity makes a rigoroustreatment impossible, which thus requires a simplification of the solutionmethods. This is why industrial simplification sometimes consists of hand-waving estimates or semi-empirical (quick & dirty) methods for treatingrealistic systems (e.g., the Steenbeck principle for electric arc physics; cf.Sect. 3.5).

Industrial R&D is characterized by limited resources (information, time,money, manpower ...). Not to afford a deep scientific investigation canbe another reason to use efficient quick & dirty approaches. Neverthe-less, a rough estimate is better than nothing - unless the result is wrong.Sometimes efficient scientific methods, which balance the effort (costs) andthe accuracy of the result (benefit), can also be useful in the innovationprocess, when ideas have to be screened with respect to feasibility. 3

1R&D = research and development (German: F&E = Forschung und Entwicklung).2A. Speiser: Der theoretische Physiker in der Industrieforschung, Z. Phys. B - Condensed

Matter 68 I-V (1987).3It is a fact that only a very small fraction of all invention ideas created will take all the

hurdles and end up in a profitable product.


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CONTENTS 5

Last but not least, industrial R&D projects are strongly interdisciplinary(not only among different science areas, but involve also salesmen, con-trollers, managers, ... ) and require a very specific communication abilityof a researcher.

This course is structured roughly as follows. After briefly touching the mostrelevant notions of economic modelling, the principles of optimizing technicaldevices will be introduced. The third, and largest part of the course, containssome useful instruments from the modelling toolbox of the theoretically workingscientist in industrial R&D.

A remark on the exercises: They are not aimed as tests but rather as anopportunity for learning by doing. As in real life, the problems are not alwayssolvable only with the help of material discussed in lectures, but requires some

own thinking and researching by the students. As usual, a problem is oftenpartly solved once the problem is understood. The students are free to discussin groups (team-work) and to ask the assistant for help.

As these lecture notes are written in LaTex, which always produces a professional-looking output, the students should be aware that it is nothing but a simplescript of a university course and can contain misprints and didactic imperfec-tions. I will appreciate remarks and improvement proposals of critical students.


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6 CONTENTS


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Chapter 1

Economic Modelling

The main task of economic modelling in industry is economic decision making.Economic modelling of an object (a technical device or even a market, capitalgoods, stocks, projects, ...) requires thus an appropriate measure for its valuex. By obvious reason, x is often defined as the monetary value of the object(in units of CH F, $, EU R, etc.). Such monetary values will be considered inthe following.

A single rationally behaving actor is assumed to decide according to anoptimization of a certain goal function; this is discussed in Sect. 1.1. Section1.2 explains the goal function if the actor is an industrial company. In thecase of more than one actor, e.g. in a market, all of them having their owninterest, a single optimizing function is usually absent; this issue is related to

game theory.1

A simple example of market modelling will be discussed in Sect.1.3 in the framework of an exercise.

1.1 Utility Maximization

The task is to make a decision on which value of x out of a set X R shouldbe selected. A simple assumption is that the decision maker maximizes a utilityfunction U(x) (Nutzenfunktion). Such behavior is called rational, in contrastto irrational behavior often observed in consumer and financial markets. Behav-ior in industrial markets (german: Industrie- oder Investitionsguter Markt) isbelieved to be rational.

Properties of the utility function U

monotonously increasing: U 0, i.e., a higher value x has not a smallerutility

U is limited from above (K R: limx U(x) < K)

U characterizes the risk behavior: U < 0 risk aversion, U = 0 riskneutral, U > 0 risk searching.

1J. von Neumann, Oskar Morgenstern: Theory of Games and Economic Behavior. Prince-ton, NJ. Princeton University Press. 1944 sec.ed. 1947

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8 CHAPTER 1. ECONOMIC MODELLING

Example 1 Gambling or not?

Consider a decision maker with a concave utility function U(x) (see Fig. 1.1),and assume fixed p with 0 < p < 1. Will he prefer a certain value xp =px1 + (1 p)x2 (i.e, with probability one), or will he prefer a game, where heobtains with probability p the value x1 and with 1 p the value x2?

For both cases, the expectation value of x is xp. But the expectation of theutility is U(xp) for the certain value and pU(x1) + (1 p)U(x2) for the game.Due to the concavity of U, the utility of the game is smaller than the utility ofthe certain value, hence the game is refused. For convex U the game is favored.

Figure 1.1: Concave utility function: the utility of a certain value x is always

higher than the utility of a game with the expectation value x.

Example 2 St. Petersburg ParadoxWhat will a rational decision maker pay for playing the following coin tossinggame. The game ends when tail appears. The prize for head at throw k is 2k$.

The probability of obtaining k times head in succession is (1/2)k. If oneaccepts that the value x of a game is equal to the expectation value of the prizeof the game, then x =

1 (1/2)

k2k$ = . However, it is an empirical factthat nobody will pay more than a few dollars. Why? The answer has beengiven by Daniel Bernoulli in 1738 who introduced the utility function U. DieBerechnung des Wertes einer Sache darf nicht auf ihrem Preis basiert werden,sondern auf ihrer Nutzlichkeit. Obviously, the expectation of U remains finite

because U is limited from above:

1 (1/2)k

U(2k

)$ = finite.

We conclude that in the presence of probabilities, utility functions must beconsidered. However, in the following we restrict the considerations on risk neu-tral decision makers (U = 0), so that the monetary value x can be considered,which is a reasonable assumption for many industrial decision problems.

1.2 Net Present Value (NPV)

The NPV (Netto-Kapitalwert) of an object (e.g., an income, a technical de-vice, a project, etc.) is the total monetary value of all cash flows associated with


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1.3. MARKET MODELLING 9

this object, evaluated at a specific time. It is important to realize that according

to economics, same monetary values at different times are not necessarily equal.For example, x = 1$ now and x = 1$ in the future, say in one year, are not thesame. The reason is that lending or borrowing money is always associated witha finite interest for the creditor, or the cost of capital for the industry thatborrows money, e.g., in order to develop a technology.2 Consequently, assum-ing a yearly (technical) interest rate r (technischer Zins), 1$ in one year isworth today only 1$/(1+r). In order to make an economic decision today basedon information on costs and benefits distributed in the future (that is a timeseries of cash flows), one first has to discount all monetary values for a fixedtime, say to the value today, and then to sum them up. (Obviously, becausefuture is never certain, here probabilities come into play and utility functionsshould be considered. However, as mentioned we will assume risk neutrality).If the value accumulating during year n = 1, 2,... is xn (the sign determines

whether it is income or expenditure), the present value is

P V(x) =

n=1

xn(1 + r)n1

. (1.1)

By definition, the value at n = 0 will be reserved for values at time zero, and isassociated in most cases with investment costs. The value for n = 1 accumulatesduring the first year and is not discounted.

The word net in the expression NPV reflects the fact, that it includes allbenefits and costs that accumulate during the life of the object to be evaluated.In the framework of economic optimization of technical devices, one usuallywrites

N P V = BLC

CLC , (1.2)

where BLC and CLC are the present values of the life cycle3 benefits and life cycle

costs (with positive sign; includes operation cost, disposal, etc.), respectively.The costs are further divided in (usually unique) investment costs and in (usuallyperiodically occurring) operation costs, CL = Cinv + Cop.

Economic optimization of technical devices simply means

N P V = maximum (1.3)

with the constraint that N P V 0. Of course additional constraints can occur.The optimization criterion (1.3) is the main principle from which many proper-ties of things from single objects (e.g., design of technical devices) up to complexsystems (e.g., price formation in markets) may be understood.4 In Sects. 1.3and 2.6 these issues will be studied in more detail.

1.3 Market Modelling

The modelling ability of physicists can be useful for studying economic markets(econophysics). Here, we discuss an example from micro-economics as an ex-ercise. The importance of market modelling lies in the fact that in order to

2The cost of capital, or the associated interest rate, generally depends on various things,like the industrial sector (because of different risks) etc.

3life cycle = time period of the objects life4Sometimes, another quantity appears: the internal rate of return rIRR , which is defined

as the rate r for which by N P V = 0.


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10 CHAPTER 1. ECONOMIC MODELLING

determine the producers benefit from a technical device, one should estimate

the price of this device on the market. Price predictions are equally importantas cost estimates. The following example is, although strongly oversimplifying,of high pedagogical value.

Exercise 1. Cournot Market Equilibrium5

Consider a market with N providers of a product. The total production costs ofprovider j for the amount xj traded by him is Kj = cjxj with constant marginal

costs cj (= dKj/dxj). Assume a relation between total demand, x =N

k=1 xj,for the product and the market price p per quantity x, given by x = A/p withpositive constant A (Note that now x is not the monetary value of something,but the total amount of traded products if the price is p).

a) Understand and interpret qualitatively the demand-price relation p(x).

(What is the meaning of A? What mean dp/dx < 0 and the limits for x and x 0? Is it realistic ?)

b) Calculate the price p, the total amount traded x, and the profits Gjin the Cournot-equilibrium by maximizing the net profits Gj({xk}) (= incomeminus costs) of the providers. The Cournot equilibrium is characterized by thefact, that every provider j can only vary its own output xj (i.e., xj/xk = 0for j = k). For the case where cj c independent of j, discuss p, x, andGtot =

Nj=1 G

j as a function of N.

Assume that the total market profit, Gtot, is the driving force of the marketdevelopment. What are the consequences for N (will there be a monopoliza-tion)?

c) What are the conditions on the marginal cost cN+1 of an additional man-ufacturer N + 1 in order that he will enter the market.

d) What is the effect of the statistical variance of the cj = c + cj on the

total market profit, at constant average c = N1N

j=1 cj?

e) For cj c, what is the change of the profits Gj , when a single provider kdecreases his cost (ck = c c; to leading order in c/c)?

5Note the different meaning of the symbols from the preceding discussion.


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Chapter 2

Optimization of TechnicalDevices

A frequent task in industrial R&D concerns optimization of products or tech-nologies (an existing device, a new prototype, or just an idea sketch etc.). Op-timization means to select the best values of the design parameters that definethe device (for example, the number of cells in a battery, the permeability of amembrane, the thickness of an electrical insulator etc.). In the previous chapterwe have seen that NPV maximization is generally the appropriate optimizationstrategy, i.e., the optimum is an appropriate balance between large benefit andlow costs. For that purpose a physical model of the ob ject to be optimized mustbe given. One goal of this chapter is to learn by simple examples how such

physical models can be constructed in mathematical terms.Of course, it depends on the generality of the problem and its details whether

it is really necessary to consider the total NPV or if it is sufficient to focus on apartial quantity, like costs, irreversibilites, losses, efficiency, power output etc.In the following, we discuss some of the most relevant principles: entropy gen-eration minimization, power maximization, and the application of NPV maxi-mization. Because irreversible effects are often the main physical factor makingtechnical devices sub-optimal, we will start with them.

2.1 Entropy Production Rate

Minimization of irreversible energy dissipation is a frequent task. Irreversibili-

ties are related to entropy production. Basic courses on thermodynamic cyclesconsider usually reversible processes,1 where the entropy change after a closedprocess cycle vanishes, i.e., S = 0. According to the second law of ther-modynamics, the efficiency is maximum for a reversible process. In order tobe reversible, a process must be conducted along thermodynamic equilibriumstates, which requires infinite time for a cycle. For the Carnot machine, thisleads to the well-known Carnot-efficiency

0 = 1 T2T1

, (2.1)

1It is assumed that the reader is familiar with the basics of thermodynamics.

11


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12 CHAPTER 2. OPTIMIZATION OF TECHNICAL DEVICES

where T1,2 are the temperatures of the hot and cold heat bath, respectively. In

the best case, the available work W from a heat H is given by W = 0H anddepends on the temperatures of the involved heat reservoirs.Obviously, the power output P (work per time, dW/dt) of a reversibly run-

ning machine vanishes due to the infinitely slow cycling velocity. Real machinesand engines, however, must deliver finite power and run through their cyclewithin finite time.2 This leads to an entropy increase per cycle (S > 0), or an

Figure 2.1: a) Heat conduction between two reservoirs (heat rate Q = dH/dt).b) Heat production by energy dissipation P in a subsystem of temperature T1at ambient temperature T2.

average entropy production rate S/t. Entropy production rates, S = dS/dt,will play an important role during this course also in later chapters. We recallthat the definition of the entropy change is dS = H/T, where H is the heatchange that must be calculated along a reversible process, independent of thenature of the true process. If (irreversible) heat generation of a (mechanical,electric etc.) power P occurs, one has H = P dt, such that at (spatially andtemporally) constant T it holds S = P/T.

In this chapter it is sufficient to understand, for spatially non-constant T, Sby the follwoing two steady-state examples.3

Example 1 Heat conductionTwo heat reservoirs (Fig. 2.1 a)) at temperatures T1 T2 are connected bya heat conductor that transfers a heat per time Q. The heat change rates ofthe two reservoirs are Q1 = Q = Q2, where Q1,2 are the heat rates of thereservoirs. The total S is the sum of the two entropy production rates:

S = Q(1

T2 1

T1) . (2.2)

(It is very general that the entropy production rate is the product of generalizedcurrent (here Q) and force (here (T1)).)

2This is why this scientific discipline is sometimes called finite time thermodynamics.3Unless stated otherwise, during the course always stationary entropy production rates will

be considered.


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2.2. ENTROPY GENERATION MINIMIZATION 13

Example 2 Power production

Consider a power device (e.g., a battery discharge over a resistor) with heatingpower P, embedded in a heat bath (the ambient) at temperature T2. Thetemperature, T1, of the environment of the heat source (system 1) is generallydifferent from T2 due to the heating (see Fig. 2.1 b)). The entropy rate of thesystem at T1 is now obviously (P Q)/T1. Because at steady state, P = Q, thesame argument as in the previous example gives for the total entropy productionrate:

S =P Q

T1+

Q

T2=

P

T2. (2.3)

Keep in mind that entropy production rate times ambient temperature equalsheat production. (Question: where appears here, for instance if P is electrical

Joule power, a product of generalized current and force ?)

2.2 Entropy Generation Minimization

In the presence of heat conduction Q and/or energy dissipation P, irreversibil-ities are present, which will have a negative effect on the overall efficiency . Adesign optimization method for technical devices, which can be applied in cer-tain specific cases, consists in minimizing the irreversibilities, i.e., the entropyproduction rate. There is a large bulk of literature on this approach.4 We will

not go into details here, but mention the importance of the constraints to thisoptimization method, and consider a simple, rather artificial but illustrative,example.

Example 1 Electric current ductConsider a linear conductor of length L and surface area A that conducts anelectric current I out of a hot system at T1 into the ambient with T2. Thegoal is to design the conductor geometry (say, A for given L) such that theirreversibility due to (linear) heat loss and Joule power production is as smallas possible. The given material properties are the electric conductivity and theheat conductivity . The entropy production rate due to Joule heating is SI =LI2/(AT2), and due to heat conduction SQ = (A/L)(T1 T2)(T12 T11 ).Minimization of the total entropy production

SI +

SQ as a function of A/Lgives A/L = I/T 0. The entropy production minimum is shown in Fig.

2.2. The result is equivalent to P = 0Q, i.e., the optimum corresponds to thebalance of available power losses. The available power of the electric power Pis P, and the available power of heat power Q is 0Q.

A similar example refers to optimization of an electric insulator (carrying avoltage drop U), which is at the same time a good heat conductor that has tocool, i.e., conduct away a heat rate Q.

4cf. A. Bejan, Entropy generation minimization: The new thermodynamics of finite-sizedevices and finite-time processes, J. Appl. Phys. 79, 1191 (1996).


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Figure 2.2: Left: Electrical link between two heat reservoirs. Right: Individualand total entropy production rates of the current duct between two heat reser-voirs as a function of A/L, for the illustrative case where the optimum is atA/L = 1 (arbitrary units).

2.3 Endoreversible Thermodynamics

Endoreversible Thermodynamics provides, in the framework of technical ther-modynamics, a simple instrument to understand qualitatively complicated realthermodynamic systems. It provides a number of interesting general results oflarge pedagogical value.

Definition An endoreversible System is an irreversible thermodynamic sys-tem that consists of subsystems, which behave fully reversibly, and where irre-versible processes occur only in the connections between these reversible sub-systems.

Example 1 Endoreversible Carnot engineConsider a reversible Carnot engine (Fig. 2.3) that works between T1,c and T2,the temperatures of the incoming hot working fluid and the lower heat bath,respectively. Furthermore, we take into account the irreversibility associatedwith the heat flow5 Q = C(T1T1,c) between the real upper heat bath at T1 andthe hot working fluid at T1,c. C is a constant heat conductance. The durationt of the whole cycle is assumed to be approximately equal to the duration

of the heat transfer from the upper heat bath to the working medium at T1,c.Furthermore, Pt is the work per cycle, and Qt and Q2t are the transferredheats. Power balance implies Q = Q1 = P + Q2. The goal is to determine theoutput power P as a function of T1,c. This temperature can be influenced bythe design of the heat exchange process and is thus a design parameter. In thefollowing it is convenient to consider the efficiency = P/Q = 1 T2/T1,c asthe design parameter instead of T1,c.

From the defining equations of and 0 (Eq. (2.1) with T1 and T2) onecan eliminate T2 and obtains T1,c = (1 0)T1/(1 ). Substitution ofT1,c in

5This irreversibility can be due to finite time of contact, heat resistance in the heat ex-changer, etc..


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2.3. ENDOREVERSIBLE THERMODYNAMICS 15

Figure 2.3: Example of an endoreversible system consisting of a reversibleCarnot machine with irreversible heat conduction from the upper heat reser-voir to the working fluid.

P = C(T1 T1,c) leads to

P() = CT1(0 )

1 (2.4)

(cf. Fig. 2.4). It is obvious that for infinitely slow cycle with = 0 thepower vanishes: P(0) = 0. For zero efficiency = 0 again P(0) = 0. SolvingdP/d = 0 yields the maximum of P at = 1 1 0 =

CA = 1

T2T1

, (2.5)

which depends only on the temperature ratio of upper and lower heat bath, asthe Carnot efficiency does. CA is known as the Curzon-Ahlborn efficiency.

6

Clearly, the optimum efficiency is between CA and 0. Whether it is closer tomaximum power or maximum efficiency, can only be answered by economicalconsiderations. The following exercise illustrates a possible way.

Exercise 2. Optimization of an Endoreversible Carnot Engine

a) Derive the power-efficiency relation for the previous example with a heatconduction law Q = C(T11,c T11 ). Sketch P() and determine at powermaximum.

b) Derive an equation for the optimum efficiency opt by minimizing the totalcosts per power of the engine. The total costs are the sum of fixed investmentcosts KI and primary energy costs, aQ (a = net present costs per heat rateQ). Define and interprete a dimensionless parameter (say, ), which determines

6This result has a rather general meaning, cf. C. Van den Broeck, PRL 95, 190602 (2005).


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Figure 2.4: Power-Efficiency relation for the endoreversible Carnot engine ofthe previous figure. The question what the best efficiency value is has to be

answered by NPV maximization.

opt/0. Calculate, sketch, and discuss opt() and Popt() for the cases 0, = 1, and .

2.4 Power Maximization

Power maximization can be a reasonable strategy, for instance in some caseswhere the primary energy is for free. The technical information needed tooptimize such devices is contained in power-efficiency curves. For illustration,we add a few simple examples where they can be expressed in the form P(),

which can be maximized and yields

.

Example 1 Ideal solar thermal absorber temperatureAn absorber at T1,c receives sun radiation of temperature T1 = 5762 K, andfeeds a reversible Carnot engine connected to the lower heat bath with T2 =288 K (Fig. 2.5). The heat current is given by the Stefan-Boltzmann law, Q =ASB (T

41 T41,c), where A is the effective absorber surface and SB is the Stefan-

Boltzmann constant. In the same way as for the previous example, one candetermine the absorber temperature T1,c associated with maximum power P.Because terms T4 occur, the maximum has to be determined numerically. Oneobtains T1,c = 2443 K, associated with an efficiency

= 0.88. Of course, thisresult is not practical because there is no absorber material that can withstandsuch a temperature.

Example 2 Photovoltaic cellThe simplest form of a photovoltaic cell can be understood as a semiconduc-tor diode with a (photo) current source driven by illumination (Fig. 2.6).The total current I is the sum of the photo-current Iph and the diode cur-rent I0(exp(eU/kbT) 1) where e, kb, T, and U are the electron charge, theBoltzmann constant, the temperature, and the voltage drop, respectively. Theconstant I0 depends on details which are not relevant for our purpose. The neg-ative sign in front of the diode current takes into account that the diode voltagecounteracts to the photo-current. (The current-voltage curve in Fig. 2.6 is thusa negative diode characteristics shifted by the photo-current).


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2.4. POWER MAXIMIZATION 17

Figure 2.5: Endoreversible model of ideal solar thermal power generation. Theunavoidable irreversibility occurs due to black-body heat radiation from the sunto the absorber.

The solar cell power P = U I is then

P = (Iph I0(exp(eU/kbT) 1)) U . (2.6)P(U) is also shown in the figure and exhibits a maximum at a voltage value U

(maximum power point, MPP). 7 You may easily calculate U numerically anddiscuss the result.

Example 3 Wind PowerThe rotor of a wind turbine decelerates the air velocity from v1 to v2 by ex-tracting kinetic energy (Fig. 2.7). At the same time, the cross section of theinflowing air increases from A1 to A2 (see Fig. Fig. 2.7 a) and b)). The regionof change is confined to the vicinity of the rotor and has a width s. In thisregion, the average velocity and cross section are v and A, respectively. The airdensity will be denoted by with the same meaning of indices.

Mass conservation implies m = vA = 1v1A1 = 2v2A2. The work done

by a velocity change dv in the volume As is given by dW = mvdv =Asvdv = msdv. Because m is constant, W = sm(v2 v1). The forceis F = W/s = m(v1 v2), and thus the power

P = F v = A(v1 v2)v2 = A4

(v1 + v2)2(v1 v2) , (2.7)

7We mention that in the context of solar cells, there is another optimality relation. Onecan show that an optimum band gap width exists, where the efficiency is maximum. Thisoptimum is based on the fact that increasing the band gap increases the available energy perphoton (which is equal to the band gap), but decreases the number of exploited photons inthe radiation (because photons with energy lower than the band gap cannot create an electronhole pair). See Shockley W. and Queisser H., J. Appl. Phys. 32, 510 (1961).


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Figure 2.6: a) Simple sketch of a p-n-diode photo-cell, where electron-hole pairsare created by photons. The charge carriers diffuse to the contacts and lead toa photo-current. b) U I and U P characteristics of the photo-diode.

where we assumed v (v1 + v2)/2. In the application, v1 is given, while v2can be tuned by the rotor design and the blade adjustment. The optimumpower (P(v2)= maximum) is obtained for v

2 = v1/3, with a maximum value of

(16/27) Av31/2. The fact that the coefficient of performance is 16/27 60%(the maximum one can capture from wind power) is called the Betz law.

Figure 2.7: a) Simple model sketch of a wind turbine with inflowing air (index 1)and out-flowing air (index 2). b) Velocity profile along the axis. The optimumpower is obtained when v2 = v1/3.

Example 4 Electrical MotorThe most simple principle of a linear DC electro-motor is sketched in Fig. 2.8a) and consist of a rod of length L, with resistance R, sliding without friction on


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2.5. RAGONE PLOTS OF ENERGY STORAGE DEVICES 19

two parallel rails with a voltage difference U, which is kept by a connection to a

battery. The occurrence of the current I through the rod and the presence of aperpendicular magnetic field B imply a force F = IBL which accelerates the rodalong the rails. The motion leads to an induced voltage Uv = BLv, where v isthe rods velocity. The current voltage relation is thus U = RI+Uv. Eliminationof I gives a relation between velocity and force: v = U/BL F R/(BL)2 (seeFig. 2.8 b)). The total power is P = IU = P0(1 w), where we abbreviatedP0 = U

2/R and w = v/vL with the idle velocity vL = U/BL. The mechanicalpower is given by Pm = F v = P0w(1 w). There is obviously a maximum ofPm at w = 1/2 (see Fig. 2.8 b)).

Because the efficiency of the motor is = Pm/P = w, one can express thepower as a function of the efficiency

Pm = P0(1

) , (2.8)

with = 1/2 at maximum power P0/4.

Figure 2.8: a) Sketch of a linear DC motor consisting of a current conducting

rod gliding along two rails in a perpendicular magnetic field. b) Velocity-forceand efficiency-power characteristics of the motor.

2.5 Ragone Plots of Energy Storage Devices

Ragone plots of energy storage devices (ESD) refer to energy-power relations,and are analogous to the above discussed efficiency-power relations.8 In orderto figure out the best ESD technology for a given application, it is important to

8T. Christen and M. W. Carlen, J. Pow. Sources 91, 210 (2000).


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know the region in the energy-power plane covered by the technologies (see Fig.

2.9). For instance, a battery for an electro-mobile has to deliver a minimumenergy (related to the minimum traveling distance between charging stops) anda minimum power (related to the needs associated with maximum accelerationand speed).

Consider an ESD initially filled with an energy E0. This energy will be usedby a load which demands for a power P. This power can be delivered for a finite

Figure 2.9: Regions of various energy storage devices in the (specific) energy-power plane. (SMES = superconducting magnetic energy storage; supercapac-itor = electric double layer capacitors with porous carbon electrodes; elco =electrolytic capacitors; film caps = e.g., metalized polymer foils).

time t, and the energy obtained by the load is E E0. The Ragone plot isdefined as the curve E(P) in the power-energy plane, i.e. each value gives theavailable energy for constant power delivery.

The following behavior is obvious for real ESD. For infinite discharge time,P 0, self-discharge (leakage) of the ESD will lead to a total loss of the storedenergy, hence E

0. The power is generally limited by internal losses to a

maximum available power Pmax (friction, internal resistance, imperfect com-bustion, etc.), where the available energy E is considerably smaller than thestored energy. Typically, there exists a maximum Emax of E in between theselimits.

The Ragone relation E(P) can be determined as follows. Let Q = (Q1, Q2,...)be the state variables of the ESD (angle coordinates, charge, current, momen-tum, etc.), which are governed by a dynamic equation Q = F(Q, t), with initialcondition Q(t = 0) = Q0. The initially stored energy is E0 = E0(Q0). Deliveryof a finite constant power P is only possible for finite time, t(Q0, P), whichhas to be calculated from the dynamic equation for the ESD. The obtained en-ergy is then given by the time-integral of the power. Because we restrict our


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2.5. RAGONE PLOTS OF ENERGY STORAGE DEVICES 21

Figure 2.10: a) An energy storage device is discharged over a load which de-mands a constant power P. b) The Ragone plot E(P) in the energy-powerplane is associated with the available energy E for the load demanding constantpower P.

considerations to constant P, the Ragone curve becomes

E(P) = P t(P) , (2.9)

where we dropped the dependence on the initial conditions.

Example 1 The Ideal BatteryThe ideal battery with capacity Q0 (inset Fig. 2.11) is characterized here bya constant (charge independent) reversible cell voltage V which depends onthe charge Q as follows: V = U0 if Q0 Q > 0 and V = 0 if Q = 0 . Ina first step, we disregard the leakage resistance RL. The power is given byP = U I = (U0 RI)I, where U0 is the terminal voltage9 and I = Q is thecurrent, and R is the terminal resistance10. The solutions of this quadraticequation are

I =U02R

U204R2

PR

. (2.10)

In the limit P 0, the two branches correspond to a discharge current I+ U0/R and to I 0. For the ideal battery, the constant power sink can alsobe parameterized by constant load resistance, RLoad. The two limits belongthen to RLoad 0 (short circuit) and RLoad (open switch), respectively.Hence we have to take the branch with the minus sign, I I, in (2.10).

The battery is empty at time t = Q0/I, where the initial charge Q0 isrelated to the initial energy by E0 = Q0U0. It is easy to include the presenceof an ohmic leakage current into the discussion. The leakage resistance RLincreases the discharge current I by U0/RL. The energy being available for the

9German: Leerlaufspannung10German: Klemmenwiderstand


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Figure 2.11: Solid curves: Ragone plots of an ideal battery with and withoutleakage (resistance RL). Dashed curve: secondary branch of the energy-powerrelation, not useful for a Ragone curve (see text). Inset: Constant power loadP connected to a battery with capacity Q0, terminal resistance R, and leakageresistance RL.

load becomes

Eb(P) = P t =2RQ0P

U0

U20 4RP + 2U0R/RL. (2.11)

Equation (2.11) corresponds to the Ragone curve of the ideal battery. In the

presence of leakage, Eb(0) = 0, and there exists a maximum at P U20 /2RRL.Without leakage (R/RL 0) the maximum energy is available for vanishinglylow power, Eb(P 0) = E0. From Eq. (2.11) one concludes that there isa maximum power, Pmax = U

20 /4R, associated with an energy E0/2 (here we

neglected a small correction due to leakage). This point is the endpoint of theRagone curve of the ideal battery, where only half of the energy is availablewhile the other half is lost at the internal resistance.

Let us finally express the Ragone plot for the battery in the dimensionlessunits eb = Eb/E0 and p = P/Pmax

eb(p) =1

2

p

1 1 p + 2R/RL . (2.12)

Ragone curves (2.12) with and without leakage are shown in Fig. 2.11. Thebranch belonging to I+ is plotted by the dashed curve.

Exercise 3: Ragone Plot of the Capacitor

In a similar way as in the previous example, determine and discuss the Ragoneplot for an ideal capacitor with capacitance C and terminal resistance R. Youcan neglect the leakage resistance. Note that discharge at constant power leadsto a nonlinear ordinary differential equation (ODE) and not to a simple alge-braic equation as for the battery. Determine from this ODE the time t andthen the Ragone curve. You have to take into account here, that the terminalvoltage U0 for the initial condition of the ODE is different from the capacitor


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2.6. ECONOMIC OPTIMIZATION 23

voltage UC,0 just before starting the discharge (UC,0 determines the initial en-

ergy E0). Is the capacitor fully discharged at t? Give an interpretation! Writethe final result in the same dimensionless form for e(p) as for the battery, andsketch e(p). Show that e(p) = 1 p gives a rough approximation.

2.6 Economic Optimization

The aim of this section is to show how technical devices should be economicallyoptimized, by applying the N P V-maximization concept of Sect. 1.2. For illus-tration, we restrict the considerations to energy storage devices (ESD).11,12

We assume that the investment costs depend on the size N and the costscN in $ per unit (e.g., volume [N] = m

3, [cN] = $/m3; mass [N] = kg, [cN] =

$/kg; or number of cells [N] = 1, [cN] = $/cell) in the form

Cinv = C0inv + cNN , (2.13)

where C0inv is a size independent part. We also assume that the same amount ofoperation costs, Cop,n, accumulates in each year n (n = 1, 2,...,) at the samerate r during the product lifetime N. The present value of the operationcosts is

Cop = C0op + ced

one year

dtPreq

, (2.14)

where C0op is energy independent, and the second part represents the cost of theprimary energy needed to charge the ESD. It depends on the power demandPreq

(t) (i.e., the power delivered by the ESD at time t to the load, [Preq

] = W),on the cost of energy ([ce] =$/J), and on the total energy efficiency (or round-trip efficiency, including also charging losses). According to the geometricalseries (1.1), the term d = ( 1)/( 1), with = 1/(1 + r), appears. Notethat d = d() is a function of the lifetime and generally depends on the operationconditions; for simplicity we will neglect this fact below.

The net present benefits are similarly given by

BLC = bed

one year

dt Preq , (2.15)

where now the benefit (price) per output energy, be ([be] =$/J), appears. Thenet present value, N P V = BLC Cinv Cop has to maximized under theconstraint N P V 0.

13

In the following, we determine the optimum size N of an ESD with the helpof the Ragone energy-power relation e(p). This will turn out to be equivalentto find the optimum working point on the Ragone curve. First, the efficiencycan be written as = ce(p), where c is the p-independent charging efficiency.

11T. Christen and C. Ohler, J. Pow. Sources 110, 107 (2001).12See also Exercise 2.13Note that NP V = 0 is principally sufficient for an entrepreneurial decision to start a

business / make an investment, because all costs including labor costs like salaries, and costof capital, etc., should be included in the NP V. However, there might be other investmentopportunities with larger NP V.


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Secondly, one can express the NP V as a function of the design variable N

by using P = Preq/N (P is defined as the specific power, while the requiredpower demand, Preq , has units of power). In dimensionless units one has p =Preq/NPmax; hence one could also use p instead of N as the variable. Becausep 1, the minimum size is Nmin = Preq/Pmax. Optimization means

N P V = Max ! (2.16)

If we disregard terms independent of the design variable (N), the N P V is

NP V(N) = bed

one year

dt Preq ced

one year

dtPreq

cNN . (2.17)

Example 1 Battery and Simplified CapacitorWe use p as the variable and assume that Preq is constant in time during the

periods of discharge. The required energy per year is Preq with utilizationtime per year. During the time period 1 year , the ESD is charged withcharging efficiency c, or stands still. The relevant (i.e., p-dependent) part ofthe NPV (2.17) is given by

N P V(p) =cNPreqPmax

( 1

e)K 1

p

(2.18)

with

K =cedPmax

cNc(2.19)

and

=bec

ce. (2.20)

The meaning of K is the ratio between the lifetime sum of energy costs formaximum power for the corresponding utilization factor and the investmentcosts. The meaning of is the ratio between the energy benefit (the energysales price) and energy costs (taking into account the charging loss). Becausewe assume that d is p-independent, maximum N P V corresponds to minimumlifecycle costs:

1

Kp+

1

e(p)= Min ! (2.21)

The only parameter remaining in this optimization problem is K. If e(p) is adecreasing function, a local minimum of Eq. (2.21) may be expected. Mini-mization of Eq. (2.21) leads to

Kp2e + e2 = 0 , (2.22)

where the prime denotes differentiation with respect to p. From Eq. (2.22) onefinds the relation between K and the optimum operation point p:

p =1

2

2 + K1

4K1 + K2

4K1 + K2 , (2.23)

for the ideal battery Eq.(2.12) without leakage. For the approximate Ragonecurve e = 1 p of a capacitor the result is simply

p =1

1 +

K. (2.24)


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2.6. ECONOMIC OPTIMIZATION 25

Figure 2.12: Optimum operation points (dots) for K = 1 on the Ragone plotfor battery (solid) and capacitor (dashed); leakage is neglected.

For every K there is an optimum operation point on the Ragone curve (see Fig.2.12). The relations p(K) and e(p(K)) can now be discussed in the same way asdone in Exercise 2. If investment costs dominate operation costs (K 1), theoptimum operation point is at maximum power (p 1), and the size N shouldbe as small as possible (N Nmin). In the other limit K , the efficiencyhas to be maximized. We will not re-iterate details here, but rather refer to thetable in Fig. 2.13 for illustrative examples. We assume = 2000 h utilization

time per year, energy costs ce = 0.1$/kWh, and a yearly cost of capital ofr = 10%. Ni-MeH and Ni-Cd batteries are similar and differ mainly in lifetimeand efficiency. It turns out that the lead-acid battery corresponds to the highestK value among the batteries, implying that it has to be operated at a higherefficiency than the Ni-Cd or the Ni-MeH batteries. This is mainly due to thelow investment costs of the lead-acid cell, which is the far most mature batterytechnology. As a consequence, it has the lowest total energy costs among thebatteries in the table.

Although, in our simple example, the parameter in Eq. (2.20) does notappear in the function to be optimized, it appears in the constraint N P V > 0.Therefore, there exists a minimum value of the price per energy, be, below whichthe ESD leads to financial loss. The corresponding values are also listed in thetable.

The most simple battery-optimization case was treated above. There aremany extensions of the approach, including p-dependent lifetime, complex powerdemand profiles Preq(t), etc. A practically relevant exercise is to repeat the

optimization for a size dependent power Preq = P(0)req(1 + N), where P

(0)req and

are given constants. This case occurs, for instance, for battery driven electricvehicles that must carry their battery.


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Figure 2.13: Comparison of optimization results for 3 different batteries and asuper-capacitor. bmine indicates the minimum energy price required for positiveN P V (for negligible C0op,inv).


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Chapter 3

Modelling Concepts andMethods

Although some of the methods in this chapter have deep mathematical back-ground and deserve a rigorous treatment, we will not go into mathematicaldetails but remain on the surface. As in the previous chapters, the goal is toprovide an illustrative recipe list, while the interested student has to dig in theappropriate literature to gain a deeper insight. In particular, we will assumethat all mathematical objects, like scalar products, operators, derivatives, etc.are appropriately defined when needed. Furthermore, we will not introduceand justify basic mathematical concepts, although we try to motivate them byillustrative examples.

3.1 Steady States and Linear Stability Analysis

The first question to be answered when a physical, chemical, technical, etc.system is investigated by theoretical methods concerns the state(s) adopted bythe system. This means to find all physically reasonable states and to determinetheir stability properties.

Consider a physical system described by a (maybe multi-component) statevariable1 (phase or state space ) and a dynamic equation (the state maydepend on time t)

t = F[] (3.1)

with initial conditions (t0) = 0 (and boundary conditions if Eq. (3.1) is apartial differential equation in space-time). In Eq. (3.1), (parameterspace ) is a (maybe multi-component) control parameter (parameter valuesthat can be controlled or tuned externally). Practically relevant questions are

Existence: are there solutions (states) (t) ? Uniqueness/ambiguity: how many solutions exist ? Type of states: are they steady states (i.e., time independent solutions),

time periodic states, non-periodic time dependent states (quasi-periodic,chaotic, ...) ?

1In analogy to phase transition theory, is sometimes called order parameter.

27


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28 CHAPTER 3. MODELLING CONCEPTS AND METHODS

Stability: Which states can be realized ? Are they globally stable, metastable,... ?

Are there symmetries and conserved quantities ? Are there thermodynamic equilibrium states ? Are there steady (flow)

states near thermodynamic equilibrium ?

Existence of a stationarity principle (Ljapunov functional, potential, ac-tion, entropy production rate, ...) ?

All answers to these questions are functions of . There are manifolds in defined by -parameterized solutions . The parameter space is divided inregions associated with different properties of the system , i.e., number N of

solutions (n) (n = 1,...,N), different stability properties, etc. It is thus con-

venient to discuss the solutions in a so-called bifurcation diagram in the space (i.e., (, )). Bifurcation theory describes how solutions or states of asystem appear and/or disappear when control parameters are varied. Typicalapplication examples are related to stability issues of devices, or to switchingprocesses, as will be discussed in the exercises 4 and 5, respectively. In thefollowing, we will consider some simple cases that can be graphically illustratedand illuminate the mathematical background.

Example 1a Single-component order and control parametersConsider = = R. The bifurcation diagram consists then of curves in theplane (, ) (Fig. 3.1). Equation (3.1) is an ordinary differential equation for areal function (t). The function F = V can be expressed as derivative of apotential V

() with a -dependent shape. Its optima are the steady states,

which are functions of and are denoted by (n)().(i) V = 2 3/3 = F = 2 2 = (1)() = , (2)() = . There

are two solution branches which intersect at = 0. Such a situation is calledtranscritical bifurcation.

(ii) V = + 3/3 = F = 2 = (1)() = , (2)() = .Here 0, for negative there is no steady state solution. This bifurcation iscalled saddle-node bifurcation, because at = 0 a saddle and a minimum ofthe potential merge.

(iii) V = 2/2 + 4/4 = F = 3 = If < 0, only one solutionexists: (2) = 0. For 0 there are three solutions, (1)() = , (2)() =0, (3)() =

. This bifurcation is called pitchfork bifurcation.

Having found the basic states, their stability must be determined. In theframework of linear stability analysis, one has to investigate wether an infinites-imal disturbance of the state under investigation decays or increases. Weagain focus on steady states, i.e. time-independent solutions () of F[] = 0.

Linearization of Eq. (3.1) leads to t = DF[()]. Here, DF repre-sents the first derivative (the Jacobian), and terms of higher order in areneglected. If k = k() are the eigenvalues of DF and k the associatedeigenvectors (eigen-modes), disturbances behave as k exp(kt). The in-dex k labels the eigenvalue spectrum, which can be discrete or continuous. Asusual, the spectrum is ordered according to the real parts of the eigenvalues,such that Re(k) > Re(j) for k < j. Hence, for a real discrete spectrum


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3.1. STEADY STATES AND LINEAR STABILITY ANALYSIS 29

Figure 3.1: Bifurcation diagrams for the cases discussed in Example 1. (i) Sta-bility exchange at intersection points. (ii) Stability exchange at a turning point(saddle node bifurcation). (iii) Stability exchange at a pitch-fork bifurcation.Along the solution = 0 the stability changes at the crossing point. Along theother solution branch, the stability changes twice due to the turning point andthe crossing point, hence the branch = 2 remains stable.

0 > 1 > ... > k > k+1 > .... Only if for all real parts Re(k) < 0 holds,

all disturbances will decay and the state () is stable. For the discrete realspectrum, the stability requirement is, for instance, 0 < 0. On the other hand,if at least one real part is positive, the state is unstable.

Note that the eigenvalues depend on the control parameters . If a controlparameter is varied, an instabilityis said to occur if the largest real part changesits sign from negative to positive. Linear stability analyis can thus be summa-rized as follows. Determine the spectrum of DF, i.e. solve the linear eigenvalueproblem

= DF[] . (3.2)

The state is stable if the spectrum k() lies in the left half of the complexplane. An instability is said to occur at a critical control parameter value c if

there the first eigenvalue gets a positive real part. If the dimension is largerthan one, not a critical point but rather a critical hyper-surface in exists. Onthis hyper-surface the largest real part of the eigenvalues of the stability problemvanishes; the associated critical states are called marginally stable. The associ-ated eigenvector is called the unstable mode and characterizes the physicaldisturbance which grows exponentially. Below we will see that eigenvalues canidentically vanish due to symmetries, which must be distinguished from a zeroeigenvalue at an instability.

Example 1b Single-component order and control parametersWe return to Example 1a and Fig. 3.1. Linearization yields DF = F

= V .


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Figure 3.2: The spectrum of the linear stability analysis depends on the controlparameter(s) . An instability occurs at a critical parameter c when the largest

real part of the eigenvalues becomes positive.

Hence minima of V are stable and maxima (and saddle points) are unstable.One finds for the above examples

(i) F = 2 = (1)() = 2, (2)() = 2. One concludes that (1) isunstable and (2) is stable for < 0, and they exchange their stability proper-ties at the crossing point ( = 0). In Fig. 3.1, stable and unstable states areindicated by solid and dashed curves, respectively.

(ii) F = 2 = (1)() = 2

, (2)() = 2 (recall 0). Thestability behavior of (1,2) is exchanged at the turning point, = 0.

(iii) F = 32 = For < 0, the only solution is stable, (2)() = . Forpositive , the three solutions have the following eigenvalues:

(1)

() = 2,(2)() = , (3)() = 2. Obviously, (1,3) are stable, while (2) is now un-stable at positive . Note that = 0 is both an intersection point anda turningpoint. While the solution (2) that is only intersected, changes stability, thesolution that is intersected at its turning point keeps its stability behavior.

Case (ii) and (iii) are two often occuring instabilites in practice. For instance,in case (iii), upon varying a control parameter, a formerly stable solution ((2))becomes unstable at a critical value (here c = 0). A small fluctuation willthen first exponentially increase in time and saturate eventually at (1) or (3).Which one will be selected depends on the fluctuation and is thus a question ofprobability. In this example the new state is close to the old one if is close toc. In case (ii) the behavior is different. An unstable parameter value ( < 0)

even close to the critical value (c = 0) will lead to transient that will go to astate (not contained in the model) far from the critical steady state.

In the framework of bifurcation theory it can be shown that there is a con-nection between stability change and bifurcations (e.g. at turning points andintersection of solutions), which follows from the theorem of implicit functions.At such points an stability exchange occurs. If a stability exchange occurstwice at the same point (e.g. crossing at a turning point), the stability of thecorresponding solution branch remains. There is a large zoo of different bifur-cation scenarios involving generic (structurally stable) and non-generic (struc-turally unstable) cases. Correspondingly, an extensive specialist dictionary ex-ists (subcritical, supercritical, Hopf, hard mode, soft mode, ...), which will not


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be stretched out here; we rather refer to the appropriate literature.2

The presence of symmetries usually leads to significant simplifications. Forinstance, case (iii) of example 1 had mirror symmetry . This symmetrygroup has two elements, namely the inversion and the identity .It is obvious, that any solution which breaks this symmetry (i.e., which isnot invariant if mirrored; to be concrete: = 0), can be used to create anothersolution by its mirror image. The number of elements of the group propertiesof the symmetry group are related to the number of solutions, which can begenerated by the group operations from a symmetry breaking solution. Mostimportant is the following fact. If a state () breaks a continuous symmetryof the system , then an identically vanishing eigenvalue exists, i.e., () = 0. The associated eigenvector is called Goldstone mode.

Goldstone modes are not only relevant in the context of stability analysis.

They are useful for describing and modelling the behavior of complex systemswith the help of collective coordinates, as will be discussed in a later exercise.Due to its relevance, we will now briefly explain in more detail the effect ofsymmetries and the origins of Goldstone modes.

Definition: According to Eq. (3.1), one may consider F as a velocityfield, which defines trajectories that are the solutions of Eq. (3.1) (we drophere the index ). A diffeomorphism g (i.e., bijectiv, g and g1 differentiable)is called a symmetry of F, if F is invariant under g, i.e., F[g] = gF[]. Thesymmetry maps solutions (trajectories) of (3.1) to solutions (trajectories) of thisequation. The symmetry operations of an equation form a group, the symmetrygroup.

Figure 3.3: Plane vector field F (with R2) with rotation symmetry. g is arotation with angle between 0 and 2.

Let F be symmetric under the (Lie) group {gs | s S differentiablemanifold; g0 = 1}. Examples are the group of continuous translations in d-

2See, e.g., J. D. Crawford, Introduction to bifurcation theory, Rev. Mod. Phys. 63, 1991,991.


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dimensional space, or rotations in R2 or R3. If 0, with 0 = s gs0 fors = 0 (this inequality defines symmetry breaking) is a stationary solutionof F[] = 0, then also s is a stationary solution, F[s] = 0 (s S). Inother words, if a symmetry breaking solution is known, one can create moresolutions (the symmetry manifold) by the action of the symmetry group. FromdF[s]/ds = 0 and the chain rule one obtains

DF[s]ss = 0 , (3.3)

i.e., GM ss is a Goldstone mode. The meaning of this zero eigenvalue isthat a displacement of an otherwise stable state in direction of the Goldstonemode (i.e., tangential to the symmetry manifold) does not lead to a restoringforce. As will be shown in Sect. 3.2, a weak symmetry breaking force can leadto a slow motion of the state on the symmetry manifold.

Example 2 Translational symmetryConsider F[] = f() + 2x, with x, t, and (x, t) real (and one-dimensional).If0(x) is a spatially inhomogeneous steady state, then also gs0(x) 0(xs)is a steady state for all s R. The spectrum of the linear stability problemhas an eigenvalue 0 with eigenvector 0. It is clear that = 0s isthe infinitesimal displacement of 0. The equation t(

0s) = 0 implies that a

spatial displacement of a stable solution 0(x) is not restored. In the presenceof weak noise, for instance, this can lead to a Brownian motion of s, i.e., adiffusion of the solution in x-space. The eigenvalue spectrum has the form0 = 0 > 1 > ... > k > k+1 > ....

Example 3 Generalized third law of Keppler

Appropriate elaboration of symmetry properties can provide relevant infor-mation on the behavior of a system. As an example, consider Newtons lawx(= d2x/dt2) = F with homogeneous force F(kx) = kdF(x). Consider thegroup gs acting on (x, t) and defined by gs(x, t) = (x exp(s), t exp(s)). Thisstretching transforms the acceleration as x exp(( 2)s)x, and the forceF exp(sd)F. The Newton equation remains invariant if (1 d) = 2,i.e. if this relation between and holds, gs is a continuous symmetry groups. If a particle needs a time T1 for a trajectory of length L1, then a particlemoving on the stretched trajectory with length L2 = L1 exp() needs the timeT2 = T1 exp() with = (1 d)/2. Consequently,

T2T1

= L2L1

1d

2

.

For d = 2 this gives Kepplers third law T2 L3, which is proven to be aconsequence of the specific symmetry of the force.

Exercise 4 Stability analysis for thermal runaway

Consider the one-dimensional heat conduction problem

ctT = 2xT +

U

L

2with boundary conditions xT = T at x = L/2. Here , c, , and U aremass density, specific heat, heat conductivity, and applied voltage. The con-


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ductivity is given by (T) = 0(1 + T /T0). Assume first homogeneous Dirichlet

boundary conditions, i.e., an infinite heat transfer coefficient .a) Transform all quantities and the heat equation to dimensionless units and

determine the control parameter which appears as the factor in front of thedimensionless heat source term.

b) Calculate the solution T(x), sketch the bifurcation diagram (, Tmax),and determine the critical parameter value c.

c) Perform a stability analysis by solving the associated eigenvalue problem,and determine again c. How changes the critical value c as a function of ?

(d) Determine the order of magnitude of the critical voltage and the char-acteristic time scale for a typical polymeric electric insulation material.

Exercise 5 Interruption of an electric current

If one tries to interrupt an electrical current by separating two contacts, aconductive plasma channel (electric arc) is created, the ionization of which issustained by the ohmic heating UI. Here, U and I are voltage and current ofthe arc, respectively. A simple model for the arc conductance G = I/U, whichreproduces qualitatively the arc behavior, is given by (the Mayr model)

dG

dt=

G

UI

K 1

where is a typical charge-carrier generation-recombination time, and K is thecooling power (heat radiation, convection, etc.); both parameters are assumedto be constant.

a) Give a physical interpretation of this equation. Determine the steadystates for (i) imposed current I (current control) and (ii) imposed voltage U(voltage control) and discuss their stability properties.

b) Consider a series connection of the arc, a voltage source U0, and a seriesresistance R. (i) Show with a sketch in the I-U diagram, that a saddle-node bi-furcation occurs. (ii) Which limit cases for U0 and R correspond to the cases (i)and (ii) of a)? (iii) Write the dynamic equation for the unitless variable g = RG,

determine the relevant control parameter , find all steady state solutions, anddiscuss the bifurcation diagram in the g plane. What happens at currentinterruption when the driving voltage U0 is lowered?

c) Discuss interruption of an alternating current near the zero crossing forlarge R and large U0, with U0 = U0 sin(t), by linearizing the voltage in time andcomparing the conductance decay time and the traversal time of the monostableU0-region (near t = 0).

3

3This problem can also be analytically calculated under the assumption, that 1, andthat for t < 0 I = It is imposed (current control), and for t > 0 the voltage is imposed(voltage control).


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3.2 Collective Coordinates

If the stable state of a system breakes a continuous symmetry, there existsa manifold of non-equivalent solutions described by the parameter s. Often,physical systems (Eq. (3.1)) can be decomposed such that F = F(0) + F(1),where F(0) is symmetric with respect to a symmetry group gs, while F

(1) is notsymmetric, but is small such that F(1) acts only as a weak disturbance. Inthat case, one can make the ansatz for the solution

= (0)s + (1)s + ... (3.4)

where (0)s is a symmetry breaking steady state of the undisturbed system, i.e.,

F(0)[(0)s ] = 0. The weak symmetry breaking force has two effects. First, it

weakly deforms the shape of (0)s . This deformation, which is described by

(1)s

in leading order, is only weak if the force is small compared to the restoringforces, which are related to the finite eigenvalues of the stability spectrum of the

undisturbed system. Secondly, it drives the state (0)s in direction of the Gold-

stone mode, where the restoring force vanishes. In other words, the symmetryparameter s becomes a slow dynamic variable and is thus time dependent,s = s(t). Slow means, that the velocity s is of order . Insertion of the expansionin Eq. (3.1) yields

ss + t(1)s = DF[

(0)s ]

(1)s + F

(1)[(0)s ] + O(2) .

In lowest order of , rearrangement gives

DF[(0)s ](1)s t(1)s = ss F(1)[(0)s ] . (3.5)

The previous equation can be understood as a linear inhomogeneous equation

for (1)s of the form A

(1)s = h. There exists a solvability condition (Fredholm

alternative) which requires that h is perpendicular to the kernel of the adjointA. In our context, this means that scalar product of the right hand side of Eq.(3.5) and the (adjoint) Goldstone mode vanishes.

Figure 3.4: Symmetry manifold (0)s with direction of Goldstone mode s

(0)s

and perpendicular direction t(1)s .

The geometrical meaning is that the dynamic equation (3.1) is projected ontothe symmetry manifold of F(0). The result is an ordinary differential equation


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3.3. NONEQUILIBRIUM SYSTEMS 35

for the collective coordinate s given by

s = < ss | F(1) >< ss | ss > . (3.6)

The following exercise should clarify these issues.

Exercise 6 One-dimensional domain wall subject to a weak force

Consider a 1-d system (space coordinate x) with state variable (x, t), andboundary conditions (x ) = 1 and (x ) = 1. Assume F[] =2x dW/d + f with W() = 2(2 2), and f(, x) is a weak disturbancewhich breaks the symmetry of the unperturbed system.

a) What are the symmetries of the unperturbed system. Sketch a steady

state (x, s) of the undisturbed system (f = 0) (Hint: use the analogy of thesteady state equation with Newtons equation for coordinate and timex). Discuss the stability of the solution, and show that the zero eigenvalue cor-responds to the translation symmetry (Hint: for the stability spectrum, use theanalogy to the Schrodinger equation).

b) Derive the equation for the domain wall motion in the form s = dH(s)/dsfrom Eq. (3.6).

c) Sketch H(s) for f = vx a(x), where (x) is the Dirac delta func-tion. Discuss the behavior of the domain wall for a > 0 and v > 0. What is thecondition for a stable steady state?

Exercise 7 One-dimensional domain wall between metastable and

stable states

In the previous exercise, the potential was symmetric such that the two stablestates had the same energy. Determine the domain wall velocity between twobistable states 1 and 2 with energy difference W = W(1) W(2) (butnow f 0).

3.3 Nonequilibrium Systems

States are often categorized with respect to their distance to thermodynamicequilibrium. In particular, one distinguishes between equilibrium states, which

are characterized by maximum entropy, and nonequilibrium states, which exhibitgradients in thermodynamic potentials, finite currents, entropy production etc.Nonequilibrium states are further divided into weak nonequilibrium (or near

equilibrium) states, where the deviations from equilibrium can be treated inthe framework of linear response theory, and strong nonequilibrium(or far fromequilibrium) states.

For systems near equilibrium, linear relations between generalized currentsJk and generalized forces Xn exist in the form

Jk =n

LknXn (3.7)


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where Lkn are the Onsager coefficients. Simple examples of generalized forces

and currents have already been mentioned after Eqs. (2.2) and (2.3). There, weconsidered single currents that are each related to a single transport coefficient.A simultaneously applied temperature difference T = T1 T2 and electricvoltage4 V = V1V2 lead to a heat current J1 = L11X1+L12X2 and an electriccurrent J2 = L21X1 + L22X2 with X1 = T

12 T11 and X2 = V /T. The

non-diagonal elements Lkl indicate electro-thermal and thermo-electric cross-effects. The Onsager coefficients Ljk as such are phenomenological constants,which have to be determined by either a basic theory or measurements. For theelectro-thermal case, the four Onsager coefficients are related to the electric andheat conductances and the Thomson and Peltier coefficients.

The entropy production rate S can be expressed as

S = n

JnXn . (3.8)

(cf. Eqs. (2.2) and (2.3)).The reader is invited to consult standard textbooks5 for details on linear

nonequilibrium thermodynamics and to learn about the properties of Lkn. Wemention here only the three most important issues:

The second law of thermodynamics, S 0, implies that the matrix Lknis positive semi-definite (or positive definite).

Micro-reversibility (time-reversal symmetry of the microscopic equations)implies the Onsager-Casimir relations Lkn(B) = knLnk(B), wherek =

1 if the associated force (Xk) is even/odd under time reversal, and

B stands for all the state variables that change sign under time reversal(e.g., the magnetic field). As an example, remember the conductancematrix for the Hall effect.

The principle of Curie says that only currents and forces associated withtensors of the same order are linked together. This leads to a restrictionof nontrivial Onsager coefficients.

In Sect. 2.1, S was used for design optimization of technical devices, Oneoption was to minimize S as a function of design parameters. In the followingsection, Swill become important in a different context. Namely, it will be shownthat in certain cases physical processes behave as if they optimize the entropyproduction rate. This behavior can be very helpful for modelling systems in

cases of restricted resources.

Prior, however, we mention that nonequilibrium states can further be de-composed in cases, where local thermodynamic equilibrium (LTE) holds, andcases where local nonequilibrium prevails. LTE applies if the thermodynamicrelations (e.g., caloric and thermal equations of state of a medium) hold locally,i.e., in infinitesimal volume elements. An example is turbulent fluid flow in hy-drodynamic systems, described by the Navier Stokes equation or gas dynamic

4More concrete, one must consider the electrochemical potential.5H. J. Kreutzer, Nonequilibrium thermodynamics and its statistical foundations, Clarendon

Press, Oxford, 1981.


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3.4. OPTIMIZATION PRINCIPLES NEAR EQUILIBRIUM 37

equations. Although globally the turbulent state is far from equilibrium, lo-

cal equilibrium still holds, i.e., the velocity distribution of the molecules is anequilibrium Maxwell distribution such that a local temperature can be defined.A necessary condition for local equilibrium is that length and time scales ofthe microscopic equilibration processes (e.g., scattering) are negligibly short ascompared to the scales of interest.

On the other hand, local nonequilibrium occurs if the equilibration processesare ineffective on the scales of interest (at interfaces, surfaces or in rarefied (verydilute) gases, electric electrode falls in electric arcs, ...), or in the cases of micro-instabilities, where the Maxwell distribution in velocity space is unstable (whichcan occur, e.g., in plasmas). The length scale at which the gas equilibrates iscalled the Knudsen length; the nonequilibrium layer in a gas at a solid surfacewithin a Knudsen length is called the Knudsen layer (see Example 3 in Sect.3.5 ).

3.4 Optimization Principles near Equilibrium

Due to the characteristics of industrial physics mentioned in the introduction,it would be advantageous to have a simple optimization principle for modellingnonequilibrium, dissipative, irreversible systems. Unfortunately, a general vari-ational principle for macroscopic nonequilibrium systems does not exist6. Butoptimization principles hold near equilibrium, if Ljn is symmetric. The followingthree principles can be found in the literature:7

Principle of least dissipation (Onsagers principle)Consider Ljk symmetric and positive definite, such that L

1jk := RjkT

1

exists and define the dissipation function

=1

2T

kl

RklJkJl .

The weak nonequilibrium state (3.7) is the optimum of ({Jk}) = S with S from Eq. (3.8), which proves the existence of an optimizationprinciple.

Principle of minimum entropy production rate (MinEP; Prigogines prin-ciple)It states that the entropy production rate of a stationary process becomes

a minimum under certain constraints. It is nothing but an alternative formof Onsagers principle for stationary processes (cf. Ichiyanagi, Sec. 3.3).A discussion can be found in many textbooks on linear nonequilibriumthermodynamics (see also Example 1 below).

Principle of maximum entropy production rate (MaxEP)It states that a closed, isolated system in a nonequilibrium state equili-brates in way where it maximizes the entropy production under certain

6Even the equation F(x) = 0 can in general not be transformed to an optimization problemV(x ) with a scalar function V.

7M. Ichiyanagi, Physics Reports 243, 125 (1994); L. M. Martyushev and V. D. Seleznev,Physics Reports 426, 1 (2006).


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constraints. Near equilibrium, MaxEP is also related to the two previ-

ously mentioned principles. In the general case, MaxEP is not valid butcan often be a good approximation, and is thus useful for modeling com-plex dissipative systems and irreversible processes in the case of lack ofresources. In the following we will mainly focus on MaxEP.

Figure 3.5: A (closed and isolated) system in a nonequilibrium state x = 0aims to evolve towards the equilibrium state x = 0 at maximum entropy. Themaximum entropy production principle assumes that this equilibration occurs,under certain conditions, by maximizing S.

A hand-waving physical explanation of MaxEP goes as follows. A dissi-pative system in a nonequilibrium state aims to equilibrate by an irreversible

process, ending up in a state with maximum entropy (see Fig. 3.5), i.e., inthe most probable state. What drives this equilibration process is the micro-scopic thermal motion, i.e., the fluctuating degrees of freedom. They act onfinite but very short microscopic time and length scales. From a macroscopicpoint of view most types of nonequilibrium unbalances go immediately toequilibrium. However, there are constraints due to conservation laws (energy,momentum, mass, ...), which must be satisfied during equilibration. They arerepresented by continuity equations. As a consequence, these variables are slow.For instance, an inner energy unbalance in the form of constant energy densityexcept of a very localized disturbance leads to a slow heat diffusion process.

In summary: MaxEP assumes that a nonequilibrium system aims to maxi-mize the entropy production rate S as fast as it can. ... as it can refers to the

constraints.

Consider Fig. 3.6 where tot inside the region bounded by the dashed curveconsists of a battery with constant voltage U and constant resistance R0, andthe general subsystem of interest . The system tot and its environment build the total isolated system, which has maximum entropy at equilibrium. Anonequilibrium situation is created by energy transfer from to the energy stor-age device, i.e. by charging up the battery. It is assumed that is so large thatchanges of its constant temperature Tamb and of its constant electrical potential(e.g., ground) due to charging and discharging of the battery are negligible. Theboundary tot of tot (dashed curve) is chosen so that it is at ground potential


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Figure 3.6: An isolated nonequilibrium system, consisting of a part tot withvariable temperature T(r ) inside the dashed boundary (containing a batterywith voltage U, a constant resistor R0, and an electrical sub-system ), and its

environment (which is at constant potential (ground) and acts as a heat bathat temperature Tamb).

and at ambient temperature, T(tot) = Tamb. The part is much larger thanthe part and acts as equilibrium heat bath.

In the following, we consider only the nonequilibrium steady state of dis-charge. The power delivered by the battery is dissipated in the resistor R0 andin the subsystem . The corresponding Joule heat eventually flows to the envi-ronment . The temperature T inside the system tot is generally dependenton space and can be different from Tamb. The total entropy production rate is(see Eq. (2.3))

S =P

Tamb , (3.9)

with the power

P = R0I20 + P . (3.10)

Here I0 I is the current delivered by the battery and P is the power of .Power balance, as a constraint, reads

P = U I . (3.11)

Although MaxEP seems to be contradictory to MinEP, it is not. Example1 below should clarify that the kind of extremum depends on the convexityproperties of the function to be optimized and the constraints. Example 2 will

illustrate a microscopic way of looking at MaxEP.

Example 1 Kirchoffs law from MinEP and MaxEPLet be two resistors R1 and R2 in parallel. How are the currents I1 and I2through them distributed?

MinEP assumes as constraint a given total current, I = I1+I2. Because Tambis fixed, entropy production minimization is equivalent to power minimization,i.e., P = R1I

21 + R2(I I1)2 is to be minimized as a function of I1. This

immediately yields the correct result I1 = IR2/(R1 + R2). An illustration isgiven in Fig. 3.7

For MaxEP, one must consider the total system, where not I = I1 + I2 is


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Figure 3.7: Difference between MaxEP and MinEP. S has a minimum at theorigin of the I1-I2-plane. Along the MinEP constraint S has a minimum, whilealong the MaxEP constraints it has a maximum.

fixed but U. The constraint is thus total power balance, Eq. (3.11), which readsafter completing the square8 (for simplicity, assume R0 = 0)

R1I1 U

2R1

2+

R2I2 U

2R2

2=

U2

4

1

R1+

1

R2

.

This defines an ellipse in the plane II -I2. Maximization of S, or of P, on thiscurve is illustrated in Fig. 3.7. It coincides with the minimum of S along the

constraint with fixed current. It is now clear, that the question whether a stateis a maximum or a minimum of the entropy production rate depends on theconstraints.9

Example 2 Estimate of the universal conductance quantumThe elementary conductance quantum, e2/h (e is the electron charge and his the Planck constant) occurs in various systems, as in quantum point con-tacts or in the quantum Hall effect, where its inverse h/e2 is called the vonKlitzing constant. Consider two electron reservoirs (Fig. 3.8), which differ involtage by U, as is indicated in the upper part of the figure by the higherelectro-chemical potential (= band bottom potential plus Fermi energy) in theleft contact. Electrons in the gap between the two electro-chemical potentials

(Fermi levels) travel from the left reservoir through the ballistic 1-d channel,and arrive at the right reservoir with the excess energy eU. Because (inelas-tic) scattering is absent, there is no power loss and no entropy production inthe channel. Eventually, each electron releases its energy to the reservoir andproduces heat power eU/t, where t is the time required for this equilibrationprocess. Within this time interval, for the entropy production of an electronholds TambSel = eU/t. Here t can be interpreted as the time used for the de-cay of the excited state with energy width eU formed by electron and reservoir.MaxEP implies now that the average lifetime t is as small as possible. If the

8Quadratische Erganzung9It has to do with the curvatures of the curves in Fig. 3.7.


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lower bound for t is given by the energy-time uncertainty relation, eUt h,one obtains the maximum

Sel (eU)

2

/hTamb. This is only an order of mag-nitude estimate, because we neglected factors of order unity in the uncertaintyrelation. Each single electron produces this amount of entropy during the timeinterval t (see Fig. 3.8, lower part). The entropy production on a large timescale is obtained by averaging over all electrons contributing to the current flow.If we first neglect spin degeneracy, the Pauli exclusion principle implies that ev-ery electron must come alone, i.e., the Sel-pulses in the lower part of Fig. 3.8 donot overlap. But in order to maximize the total entropy production, the elec-trons will follow each other as close as possible, so that the separation betweenthe pulses vanishes, i.e., t = 0. Consequently,

Smax e2(U)2

hTamb. (3.12)

Comparing this with the well-known relation for the Joule power, TambS =G(U)2, where G is the conductance, one concludes that G e2/h. If one hasto include spin degeneracy, a factor 2 must be added. In the presence of elasticscattering, a transmission factor can be added, and the total conductance incase of many independent channels will be given by a sum over all single channelconductivities. Other but similar conductance quanta for fermionic carriers, likethe spin conductance in anti-ferromagnetic spin 1/2 chains could be estimatedin a similar way.

Figure 3.8: Top: Sketch of the potential in a quasi-1d ballistic electron con-ductor, sandwiched by left and right contacts (electron reservoirs with Fermienergies EF), which differ in their electro-chemical potentials by eU. Elec-trons which arrive in the right contact equilibrate by releasing the energy eUin form of heat. Bottom: Each electron contributes to entropy production witheU/(Tambt). The electrons arrive with time differences t. In the text itis argued that the entropy production rate is maximized if t h/eU andt = 0.)

There exist many further examples. Important work has been done byKohler10 who has shown that the transport coefficients of the hydrodynamic

10Kohler M, Z. Physik 124, 772 (1948).


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equations associated with a linearized Boltzmann equation can be determined

by the MaxEP (and equivalently the MinEP) principle. A consequence isthe successful application of, e.g., MinEP, to radiation heat transfer withina photo-hydrodynamic framework.11 Thermal diffusion can also be shown toobey MaxEP.12

3.5 Optimization Principles far from Equilibrium

Far from equilibrium such principles are generally not valid, and if the completephysical information is given (all physical laws and parameters etc. are known),they are even obsolete. However, if some information is missing, they can some-times provide useful approximate results. The accuracy of the results dependson the quality of the given information, which is used as constraints.13 Let us

restate MaxEP: The state of an irreversible, closed, isolated (macroscopic) sys-tem can sometimes be approximated by maximizing the entropy production rateS subject to the constraints associated with the given information.. We will focusfirst on electrical systems with nonlinear current-voltage relations.14

Exercise 8 Steenbecks Principle from MaxEP

The Steenbeck principle states that at given current I, a conducting system (in Fig. 3.6, e.g., an electric arc) tries to minimize its voltage drop. Assume amodel for in the form U = U(I, z) with unknown model parameter z.

a) Show that the entropy production rate for the total system is

zS =U

R0 + IUzU

Tamb , (3.13)

MaxEP requires zS = 0, i.e., zU = 0 and positive second derivative

(2z S)zS=0 =U

R0 + IU

2zUTamb

zU=0

. (3.14)

b) Discuss the stability conditions of t

application of theoretical physics methods in industrial r&d lecture_2012

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